Theorem rpnnen1lem5 13020

Description: Lemma for rpnnen1 13022. (Contributed by Mario Carneiro, 12-May-2013.) (Revised by NM, 13-Aug-2021.) (Proof modification is discouraged.)

Hypotheses

Ref	Expression
rpnnen1lem.1	⊢ 𝑇 = {𝑛 ∈ ℤ ∣ (𝑛 / 𝑘) < 𝑥}
rpnnen1lem.2	⊢ 𝐹 = (𝑥 ∈ ℝ ↦ (𝑘 ∈ ℕ ↦ (sup(𝑇, ℝ, < ) / 𝑘)))
rpnnen1lem.n	⊢ ℕ ∈ V
rpnnen1lem.q	⊢ ℚ ∈ V

Assertion

Ref	Expression
rpnnen1lem5	⊢ (𝑥 ∈ ℝ → sup(ran (𝐹‘𝑥), ℝ, < ) = 𝑥)

Distinct variable groups:   𝑘,𝐹,𝑛,𝑥   𝑇,𝑛

Allowed substitution hints:   𝑇(𝑥,𝑘)

Proof of Theorem rpnnen1lem5

Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		rpnnen1lem.1	. . . 4 ⊢ 𝑇 = {𝑛 ∈ ℤ ∣ (𝑛 / 𝑘) < 𝑥}
2		rpnnen1lem.2	. . . 4 ⊢ 𝐹 = (𝑥 ∈ ℝ ↦ (𝑘 ∈ ℕ ↦ (sup(𝑇, ℝ, < ) / 𝑘)))
3		rpnnen1lem.n	. . . 4 ⊢ ℕ ∈ V
4		rpnnen1lem.q	. . . 4 ⊢ ℚ ∈ V
5	1, 2, 3, 4	rpnnen1lem3 13018	. . 3 ⊢ (𝑥 ∈ ℝ → ∀𝑛 ∈ ran (𝐹‘𝑥)𝑛 ≤ 𝑥)
6	1, 2, 3, 4	rpnnen1lem1 13017	. . . . . 6 ⊢ (𝑥 ∈ ℝ → (𝐹‘𝑥) ∈ (ℚ ↑m ℕ))
7	4, 3	elmap 8909	. . . . . 6 ⊢ ((𝐹‘𝑥) ∈ (ℚ ↑m ℕ) ↔ (𝐹‘𝑥):ℕ⟶ℚ)
8	6, 7	sylib 218	. . . . 5 ⊢ (𝑥 ∈ ℝ → (𝐹‘𝑥):ℕ⟶ℚ)
9		frn 6743	. . . . . 6 ⊢ ((𝐹‘𝑥):ℕ⟶ℚ → ran (𝐹‘𝑥) ⊆ ℚ)
10		qssre 12998	. . . . . 6 ⊢ ℚ ⊆ ℝ
11	9, 10	sstrdi 4007	. . . . 5 ⊢ ((𝐹‘𝑥):ℕ⟶ℚ → ran (𝐹‘𝑥) ⊆ ℝ)
12	8, 11	syl 17	. . . 4 ⊢ (𝑥 ∈ ℝ → ran (𝐹‘𝑥) ⊆ ℝ)
13		1nn 12274	. . . . . . . 8 ⊢ 1 ∈ ℕ
14	13	ne0ii 4349	. . . . . . 7 ⊢ ℕ ≠ ∅
15		fdm 6745	. . . . . . . 8 ⊢ ((𝐹‘𝑥):ℕ⟶ℚ → dom (𝐹‘𝑥) = ℕ)
16	15	neeq1d 2997	. . . . . . 7 ⊢ ((𝐹‘𝑥):ℕ⟶ℚ → (dom (𝐹‘𝑥) ≠ ∅ ↔ ℕ ≠ ∅))
17	14, 16	mpbiri 258	. . . . . 6 ⊢ ((𝐹‘𝑥):ℕ⟶ℚ → dom (𝐹‘𝑥) ≠ ∅)
18		dm0rn0 5937	. . . . . . 7 ⊢ (dom (𝐹‘𝑥) = ∅ ↔ ran (𝐹‘𝑥) = ∅)
19	18	necon3bii 2990	. . . . . 6 ⊢ (dom (𝐹‘𝑥) ≠ ∅ ↔ ran (𝐹‘𝑥) ≠ ∅)
20	17, 19	sylib 218	. . . . 5 ⊢ ((𝐹‘𝑥):ℕ⟶ℚ → ran (𝐹‘𝑥) ≠ ∅)
21	8, 20	syl 17	. . . 4 ⊢ (𝑥 ∈ ℝ → ran (𝐹‘𝑥) ≠ ∅)
22		breq2 5151	. . . . . . 7 ⊢ (𝑦 = 𝑥 → (𝑛 ≤ 𝑦 ↔ 𝑛 ≤ 𝑥))
23	22	ralbidv 3175	. . . . . 6 ⊢ (𝑦 = 𝑥 → (∀𝑛 ∈ ran (𝐹‘𝑥)𝑛 ≤ 𝑦 ↔ ∀𝑛 ∈ ran (𝐹‘𝑥)𝑛 ≤ 𝑥))
24	23	rspcev 3621	. . . . 5 ⊢ ((𝑥 ∈ ℝ ∧ ∀𝑛 ∈ ran (𝐹‘𝑥)𝑛 ≤ 𝑥) → ∃𝑦 ∈ ℝ ∀𝑛 ∈ ran (𝐹‘𝑥)𝑛 ≤ 𝑦)
25	5, 24	mpdan 687	. . . 4 ⊢ (𝑥 ∈ ℝ → ∃𝑦 ∈ ℝ ∀𝑛 ∈ ran (𝐹‘𝑥)𝑛 ≤ 𝑦)
26		id 22	. . . 4 ⊢ (𝑥 ∈ ℝ → 𝑥 ∈ ℝ)
27		suprleub 12231	. . . 4 ⊢ (((ran (𝐹‘𝑥) ⊆ ℝ ∧ ran (𝐹‘𝑥) ≠ ∅ ∧ ∃𝑦 ∈ ℝ ∀𝑛 ∈ ran (𝐹‘𝑥)𝑛 ≤ 𝑦) ∧ 𝑥 ∈ ℝ) → (sup(ran (𝐹‘𝑥), ℝ, < ) ≤ 𝑥 ↔ ∀𝑛 ∈ ran (𝐹‘𝑥)𝑛 ≤ 𝑥))
28	12, 21, 25, 26, 27	syl31anc 1372	. . 3 ⊢ (𝑥 ∈ ℝ → (sup(ran (𝐹‘𝑥), ℝ, < ) ≤ 𝑥 ↔ ∀𝑛 ∈ ran (𝐹‘𝑥)𝑛 ≤ 𝑥))
29	5, 28	mpbird 257	. 2 ⊢ (𝑥 ∈ ℝ → sup(ran (𝐹‘𝑥), ℝ, < ) ≤ 𝑥)
30	1, 2, 3, 4	rpnnen1lem4 13019	. . . . . . . . 9 ⊢ (𝑥 ∈ ℝ → sup(ran (𝐹‘𝑥), ℝ, < ) ∈ ℝ)
31		resubcl 11570	. . . . . . . . 9 ⊢ ((𝑥 ∈ ℝ ∧ sup(ran (𝐹‘𝑥), ℝ, < ) ∈ ℝ) → (𝑥 − sup(ran (𝐹‘𝑥), ℝ, < )) ∈ ℝ)
32	30, 31	mpdan 687	. . . . . . . 8 ⊢ (𝑥 ∈ ℝ → (𝑥 − sup(ran (𝐹‘𝑥), ℝ, < )) ∈ ℝ)
33	32	adantr 480	. . . . . . 7 ⊢ ((𝑥 ∈ ℝ ∧ sup(ran (𝐹‘𝑥), ℝ, < ) < 𝑥) → (𝑥 − sup(ran (𝐹‘𝑥), ℝ, < )) ∈ ℝ)
34		posdif 11753	. . . . . . . . . 10 ⊢ ((sup(ran (𝐹‘𝑥), ℝ, < ) ∈ ℝ ∧ 𝑥 ∈ ℝ) → (sup(ran (𝐹‘𝑥), ℝ, < ) < 𝑥 ↔ 0 < (𝑥 − sup(ran (𝐹‘𝑥), ℝ, < ))))
35	30, 34	mpancom 688	. . . . . . . . 9 ⊢ (𝑥 ∈ ℝ → (sup(ran (𝐹‘𝑥), ℝ, < ) < 𝑥 ↔ 0 < (𝑥 − sup(ran (𝐹‘𝑥), ℝ, < ))))
36	35	biimpa 476	. . . . . . . 8 ⊢ ((𝑥 ∈ ℝ ∧ sup(ran (𝐹‘𝑥), ℝ, < ) < 𝑥) → 0 < (𝑥 − sup(ran (𝐹‘𝑥), ℝ, < )))
37	36	gt0ne0d 11824	. . . . . . 7 ⊢ ((𝑥 ∈ ℝ ∧ sup(ran (𝐹‘𝑥), ℝ, < ) < 𝑥) → (𝑥 − sup(ran (𝐹‘𝑥), ℝ, < )) ≠ 0)
38	33, 37	rereccld 12091	. . . . . 6 ⊢ ((𝑥 ∈ ℝ ∧ sup(ran (𝐹‘𝑥), ℝ, < ) < 𝑥) → (1 / (𝑥 − sup(ran (𝐹‘𝑥), ℝ, < ))) ∈ ℝ)
39		arch 12520	. . . . . 6 ⊢ ((1 / (𝑥 − sup(ran (𝐹‘𝑥), ℝ, < ))) ∈ ℝ → ∃𝑘 ∈ ℕ (1 / (𝑥 − sup(ran (𝐹‘𝑥), ℝ, < ))) < 𝑘)
40	38, 39	syl 17	. . . . 5 ⊢ ((𝑥 ∈ ℝ ∧ sup(ran (𝐹‘𝑥), ℝ, < ) < 𝑥) → ∃𝑘 ∈ ℕ (1 / (𝑥 − sup(ran (𝐹‘𝑥), ℝ, < ))) < 𝑘)
41	40	ex 412	. . . 4 ⊢ (𝑥 ∈ ℝ → (sup(ran (𝐹‘𝑥), ℝ, < ) < 𝑥 → ∃𝑘 ∈ ℕ (1 / (𝑥 − sup(ran (𝐹‘𝑥), ℝ, < ))) < 𝑘))
42	1, 2	rpnnen1lem2 13016	. . . . . . . . 9 ⊢ ((𝑥 ∈ ℝ ∧ 𝑘 ∈ ℕ) → sup(𝑇, ℝ, < ) ∈ ℤ)
43	42	zred 12719	. . . . . . . 8 ⊢ ((𝑥 ∈ ℝ ∧ 𝑘 ∈ ℕ) → sup(𝑇, ℝ, < ) ∈ ℝ)
44	43	3adant3 1131	. . . . . . 7 ⊢ ((𝑥 ∈ ℝ ∧ 𝑘 ∈ ℕ ∧ (1 / (𝑥 − sup(ran (𝐹‘𝑥), ℝ, < ))) < 𝑘) → sup(𝑇, ℝ, < ) ∈ ℝ)
45	44	ltp1d 12195	. . . . . 6 ⊢ ((𝑥 ∈ ℝ ∧ 𝑘 ∈ ℕ ∧ (1 / (𝑥 − sup(ran (𝐹‘𝑥), ℝ, < ))) < 𝑘) → sup(𝑇, ℝ, < ) < (sup(𝑇, ℝ, < ) + 1))
46	33, 36	jca 511	. . . . . . . . . . . . 13 ⊢ ((𝑥 ∈ ℝ ∧ sup(ran (𝐹‘𝑥), ℝ, < ) < 𝑥) → ((𝑥 − sup(ran (𝐹‘𝑥), ℝ, < )) ∈ ℝ ∧ 0 < (𝑥 − sup(ran (𝐹‘𝑥), ℝ, < ))))
47		nnre 12270	. . . . . . . . . . . . . 14 ⊢ (𝑘 ∈ ℕ → 𝑘 ∈ ℝ)
48		nngt0 12294	. . . . . . . . . . . . . 14 ⊢ (𝑘 ∈ ℕ → 0 < 𝑘)
49	47, 48	jca 511	. . . . . . . . . . . . 13 ⊢ (𝑘 ∈ ℕ → (𝑘 ∈ ℝ ∧ 0 < 𝑘))
50		ltrec1 12152	. . . . . . . . . . . . 13 ⊢ ((((𝑥 − sup(ran (𝐹‘𝑥), ℝ, < )) ∈ ℝ ∧ 0 < (𝑥 − sup(ran (𝐹‘𝑥), ℝ, < ))) ∧ (𝑘 ∈ ℝ ∧ 0 < 𝑘)) → ((1 / (𝑥 − sup(ran (𝐹‘𝑥), ℝ, < ))) < 𝑘 ↔ (1 / 𝑘) < (𝑥 − sup(ran (𝐹‘𝑥), ℝ, < ))))
51	46, 49, 50	syl2an 596	. . . . . . . . . . . 12 ⊢ (((𝑥 ∈ ℝ ∧ sup(ran (𝐹‘𝑥), ℝ, < ) < 𝑥) ∧ 𝑘 ∈ ℕ) → ((1 / (𝑥 − sup(ran (𝐹‘𝑥), ℝ, < ))) < 𝑘 ↔ (1 / 𝑘) < (𝑥 − sup(ran (𝐹‘𝑥), ℝ, < ))))
52	30	ad2antrr 726	. . . . . . . . . . . . . 14 ⊢ (((𝑥 ∈ ℝ ∧ sup(ran (𝐹‘𝑥), ℝ, < ) < 𝑥) ∧ 𝑘 ∈ ℕ) → sup(ran (𝐹‘𝑥), ℝ, < ) ∈ ℝ)
53		nnrecre 12305	. . . . . . . . . . . . . . 15 ⊢ (𝑘 ∈ ℕ → (1 / 𝑘) ∈ ℝ)
54	53	adantl 481	. . . . . . . . . . . . . 14 ⊢ (((𝑥 ∈ ℝ ∧ sup(ran (𝐹‘𝑥), ℝ, < ) < 𝑥) ∧ 𝑘 ∈ ℕ) → (1 / 𝑘) ∈ ℝ)
55		simpll 767	. . . . . . . . . . . . . 14 ⊢ (((𝑥 ∈ ℝ ∧ sup(ran (𝐹‘𝑥), ℝ, < ) < 𝑥) ∧ 𝑘 ∈ ℕ) → 𝑥 ∈ ℝ)
56	52, 54, 55	ltaddsub2d 11861	. . . . . . . . . . . . 13 ⊢ (((𝑥 ∈ ℝ ∧ sup(ran (𝐹‘𝑥), ℝ, < ) < 𝑥) ∧ 𝑘 ∈ ℕ) → ((sup(ran (𝐹‘𝑥), ℝ, < ) + (1 / 𝑘)) < 𝑥 ↔ (1 / 𝑘) < (𝑥 − sup(ran (𝐹‘𝑥), ℝ, < ))))
57	12	adantr 480	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑥 ∈ ℝ ∧ 𝑘 ∈ ℕ) → ran (𝐹‘𝑥) ⊆ ℝ)
58		ffn 6736	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝐹‘𝑥):ℕ⟶ℚ → (𝐹‘𝑥) Fn ℕ)
59	8, 58	syl 17	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑥 ∈ ℝ → (𝐹‘𝑥) Fn ℕ)
60		fnfvelrn 7099	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝐹‘𝑥) Fn ℕ ∧ 𝑘 ∈ ℕ) → ((𝐹‘𝑥)‘𝑘) ∈ ran (𝐹‘𝑥))
61	59, 60	sylan 580	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑥 ∈ ℝ ∧ 𝑘 ∈ ℕ) → ((𝐹‘𝑥)‘𝑘) ∈ ran (𝐹‘𝑥))
62	57, 61	sseldd 3995	. . . . . . . . . . . . . . . 16 ⊢ ((𝑥 ∈ ℝ ∧ 𝑘 ∈ ℕ) → ((𝐹‘𝑥)‘𝑘) ∈ ℝ)
63	30	adantr 480	. . . . . . . . . . . . . . . 16 ⊢ ((𝑥 ∈ ℝ ∧ 𝑘 ∈ ℕ) → sup(ran (𝐹‘𝑥), ℝ, < ) ∈ ℝ)
64	53	adantl 481	. . . . . . . . . . . . . . . 16 ⊢ ((𝑥 ∈ ℝ ∧ 𝑘 ∈ ℕ) → (1 / 𝑘) ∈ ℝ)
65	12, 21, 25	3jca 1127	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑥 ∈ ℝ → (ran (𝐹‘𝑥) ⊆ ℝ ∧ ran (𝐹‘𝑥) ≠ ∅ ∧ ∃𝑦 ∈ ℝ ∀𝑛 ∈ ran (𝐹‘𝑥)𝑛 ≤ 𝑦))
66	65	adantr 480	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑥 ∈ ℝ ∧ 𝑘 ∈ ℕ) → (ran (𝐹‘𝑥) ⊆ ℝ ∧ ran (𝐹‘𝑥) ≠ ∅ ∧ ∃𝑦 ∈ ℝ ∀𝑛 ∈ ran (𝐹‘𝑥)𝑛 ≤ 𝑦))
67		suprub 12226	. . . . . . . . . . . . . . . . 17 ⊢ (((ran (𝐹‘𝑥) ⊆ ℝ ∧ ran (𝐹‘𝑥) ≠ ∅ ∧ ∃𝑦 ∈ ℝ ∀𝑛 ∈ ran (𝐹‘𝑥)𝑛 ≤ 𝑦) ∧ ((𝐹‘𝑥)‘𝑘) ∈ ran (𝐹‘𝑥)) → ((𝐹‘𝑥)‘𝑘) ≤ sup(ran (𝐹‘𝑥), ℝ, < ))
68	66, 61, 67	syl2anc 584	. . . . . . . . . . . . . . . 16 ⊢ ((𝑥 ∈ ℝ ∧ 𝑘 ∈ ℕ) → ((𝐹‘𝑥)‘𝑘) ≤ sup(ran (𝐹‘𝑥), ℝ, < ))
69	62, 63, 64, 68	leadd1dd 11874	. . . . . . . . . . . . . . 15 ⊢ ((𝑥 ∈ ℝ ∧ 𝑘 ∈ ℕ) → (((𝐹‘𝑥)‘𝑘) + (1 / 𝑘)) ≤ (sup(ran (𝐹‘𝑥), ℝ, < ) + (1 / 𝑘)))
70	62, 64	readdcld 11287	. . . . . . . . . . . . . . . 16 ⊢ ((𝑥 ∈ ℝ ∧ 𝑘 ∈ ℕ) → (((𝐹‘𝑥)‘𝑘) + (1 / 𝑘)) ∈ ℝ)
71		readdcl 11235	. . . . . . . . . . . . . . . . 17 ⊢ ((sup(ran (𝐹‘𝑥), ℝ, < ) ∈ ℝ ∧ (1 / 𝑘) ∈ ℝ) → (sup(ran (𝐹‘𝑥), ℝ, < ) + (1 / 𝑘)) ∈ ℝ)
72	30, 53, 71	syl2an 596	. . . . . . . . . . . . . . . 16 ⊢ ((𝑥 ∈ ℝ ∧ 𝑘 ∈ ℕ) → (sup(ran (𝐹‘𝑥), ℝ, < ) + (1 / 𝑘)) ∈ ℝ)
73		simpl 482	. . . . . . . . . . . . . . . 16 ⊢ ((𝑥 ∈ ℝ ∧ 𝑘 ∈ ℕ) → 𝑥 ∈ ℝ)
74		lelttr 11348	. . . . . . . . . . . . . . . . 17 ⊢ (((((𝐹‘𝑥)‘𝑘) + (1 / 𝑘)) ∈ ℝ ∧ (sup(ran (𝐹‘𝑥), ℝ, < ) + (1 / 𝑘)) ∈ ℝ ∧ 𝑥 ∈ ℝ) → (((((𝐹‘𝑥)‘𝑘) + (1 / 𝑘)) ≤ (sup(ran (𝐹‘𝑥), ℝ, < ) + (1 / 𝑘)) ∧ (sup(ran (𝐹‘𝑥), ℝ, < ) + (1 / 𝑘)) < 𝑥) → (((𝐹‘𝑥)‘𝑘) + (1 / 𝑘)) < 𝑥))
75	74	expd 415	. . . . . . . . . . . . . . . 16 ⊢ (((((𝐹‘𝑥)‘𝑘) + (1 / 𝑘)) ∈ ℝ ∧ (sup(ran (𝐹‘𝑥), ℝ, < ) + (1 / 𝑘)) ∈ ℝ ∧ 𝑥 ∈ ℝ) → ((((𝐹‘𝑥)‘𝑘) + (1 / 𝑘)) ≤ (sup(ran (𝐹‘𝑥), ℝ, < ) + (1 / 𝑘)) → ((sup(ran (𝐹‘𝑥), ℝ, < ) + (1 / 𝑘)) < 𝑥 → (((𝐹‘𝑥)‘𝑘) + (1 / 𝑘)) < 𝑥)))
76	70, 72, 73, 75	syl3anc 1370	. . . . . . . . . . . . . . 15 ⊢ ((𝑥 ∈ ℝ ∧ 𝑘 ∈ ℕ) → ((((𝐹‘𝑥)‘𝑘) + (1 / 𝑘)) ≤ (sup(ran (𝐹‘𝑥), ℝ, < ) + (1 / 𝑘)) → ((sup(ran (𝐹‘𝑥), ℝ, < ) + (1 / 𝑘)) < 𝑥 → (((𝐹‘𝑥)‘𝑘) + (1 / 𝑘)) < 𝑥)))
77	69, 76	mpd 15	. . . . . . . . . . . . . 14 ⊢ ((𝑥 ∈ ℝ ∧ 𝑘 ∈ ℕ) → ((sup(ran (𝐹‘𝑥), ℝ, < ) + (1 / 𝑘)) < 𝑥 → (((𝐹‘𝑥)‘𝑘) + (1 / 𝑘)) < 𝑥))
78	77	adantlr 715	. . . . . . . . . . . . 13 ⊢ (((𝑥 ∈ ℝ ∧ sup(ran (𝐹‘𝑥), ℝ, < ) < 𝑥) ∧ 𝑘 ∈ ℕ) → ((sup(ran (𝐹‘𝑥), ℝ, < ) + (1 / 𝑘)) < 𝑥 → (((𝐹‘𝑥)‘𝑘) + (1 / 𝑘)) < 𝑥))
79	56, 78	sylbird 260	. . . . . . . . . . . 12 ⊢ (((𝑥 ∈ ℝ ∧ sup(ran (𝐹‘𝑥), ℝ, < ) < 𝑥) ∧ 𝑘 ∈ ℕ) → ((1 / 𝑘) < (𝑥 − sup(ran (𝐹‘𝑥), ℝ, < )) → (((𝐹‘𝑥)‘𝑘) + (1 / 𝑘)) < 𝑥))
80	51, 79	sylbid 240	. . . . . . . . . . 11 ⊢ (((𝑥 ∈ ℝ ∧ sup(ran (𝐹‘𝑥), ℝ, < ) < 𝑥) ∧ 𝑘 ∈ ℕ) → ((1 / (𝑥 − sup(ran (𝐹‘𝑥), ℝ, < ))) < 𝑘 → (((𝐹‘𝑥)‘𝑘) + (1 / 𝑘)) < 𝑥))
81	42	peano2zd 12722	. . . . . . . . . . . . . . . 16 ⊢ ((𝑥 ∈ ℝ ∧ 𝑘 ∈ ℕ) → (sup(𝑇, ℝ, < ) + 1) ∈ ℤ)
82		oveq1 7437	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑛 = (sup(𝑇, ℝ, < ) + 1) → (𝑛 / 𝑘) = ((sup(𝑇, ℝ, < ) + 1) / 𝑘))
83	82	breq1d 5157	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑛 = (sup(𝑇, ℝ, < ) + 1) → ((𝑛 / 𝑘) < 𝑥 ↔ ((sup(𝑇, ℝ, < ) + 1) / 𝑘) < 𝑥))
84	83, 1	elrab2 3697	. . . . . . . . . . . . . . . . 17 ⊢ ((sup(𝑇, ℝ, < ) + 1) ∈ 𝑇 ↔ ((sup(𝑇, ℝ, < ) + 1) ∈ ℤ ∧ ((sup(𝑇, ℝ, < ) + 1) / 𝑘) < 𝑥))
85	84	biimpri 228	. . . . . . . . . . . . . . . 16 ⊢ (((sup(𝑇, ℝ, < ) + 1) ∈ ℤ ∧ ((sup(𝑇, ℝ, < ) + 1) / 𝑘) < 𝑥) → (sup(𝑇, ℝ, < ) + 1) ∈ 𝑇)
86	81, 85	sylan 580	. . . . . . . . . . . . . . 15 ⊢ (((𝑥 ∈ ℝ ∧ 𝑘 ∈ ℕ) ∧ ((sup(𝑇, ℝ, < ) + 1) / 𝑘) < 𝑥) → (sup(𝑇, ℝ, < ) + 1) ∈ 𝑇)
87		ssrab2 4089	. . . . . . . . . . . . . . . . . . . 20 ⊢ {𝑛 ∈ ℤ ∣ (𝑛 / 𝑘) < 𝑥} ⊆ ℤ
88	1, 87	eqsstri 4029	. . . . . . . . . . . . . . . . . . 19 ⊢ 𝑇 ⊆ ℤ
89		zssre 12617	. . . . . . . . . . . . . . . . . . 19 ⊢ ℤ ⊆ ℝ
90	88, 89	sstri 4004	. . . . . . . . . . . . . . . . . 18 ⊢ 𝑇 ⊆ ℝ
91	90	a1i 11	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑥 ∈ ℝ ∧ 𝑘 ∈ ℕ) → 𝑇 ⊆ ℝ)
92		remulcl 11237	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝑘 ∈ ℝ ∧ 𝑥 ∈ ℝ) → (𝑘 · 𝑥) ∈ ℝ)
93	92	ancoms 458	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑥 ∈ ℝ ∧ 𝑘 ∈ ℝ) → (𝑘 · 𝑥) ∈ ℝ)
94	47, 93	sylan2 593	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑥 ∈ ℝ ∧ 𝑘 ∈ ℕ) → (𝑘 · 𝑥) ∈ ℝ)
95		btwnz 12718	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑘 · 𝑥) ∈ ℝ → (∃𝑛 ∈ ℤ 𝑛 < (𝑘 · 𝑥) ∧ ∃𝑛 ∈ ℤ (𝑘 · 𝑥) < 𝑛))
96	95	simpld 494	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑘 · 𝑥) ∈ ℝ → ∃𝑛 ∈ ℤ 𝑛 < (𝑘 · 𝑥))
97	94, 96	syl 17	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑥 ∈ ℝ ∧ 𝑘 ∈ ℕ) → ∃𝑛 ∈ ℤ 𝑛 < (𝑘 · 𝑥))
98		zre 12614	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑛 ∈ ℤ → 𝑛 ∈ ℝ)
99	98	adantl 481	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝑥 ∈ ℝ ∧ 𝑘 ∈ ℕ) ∧ 𝑛 ∈ ℤ) → 𝑛 ∈ ℝ)
100		simpll 767	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝑥 ∈ ℝ ∧ 𝑘 ∈ ℕ) ∧ 𝑛 ∈ ℤ) → 𝑥 ∈ ℝ)
101	49	ad2antlr 727	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝑥 ∈ ℝ ∧ 𝑘 ∈ ℕ) ∧ 𝑛 ∈ ℤ) → (𝑘 ∈ ℝ ∧ 0 < 𝑘))
102		ltdivmul 12140	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑛 ∈ ℝ ∧ 𝑥 ∈ ℝ ∧ (𝑘 ∈ ℝ ∧ 0 < 𝑘)) → ((𝑛 / 𝑘) < 𝑥 ↔ 𝑛 < (𝑘 · 𝑥)))
103	99, 100, 101, 102	syl3anc 1370	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝑥 ∈ ℝ ∧ 𝑘 ∈ ℕ) ∧ 𝑛 ∈ ℤ) → ((𝑛 / 𝑘) < 𝑥 ↔ 𝑛 < (𝑘 · 𝑥)))
104	103	rexbidva 3174	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑥 ∈ ℝ ∧ 𝑘 ∈ ℕ) → (∃𝑛 ∈ ℤ (𝑛 / 𝑘) < 𝑥 ↔ ∃𝑛 ∈ ℤ 𝑛 < (𝑘 · 𝑥)))
105	97, 104	mpbird 257	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑥 ∈ ℝ ∧ 𝑘 ∈ ℕ) → ∃𝑛 ∈ ℤ (𝑛 / 𝑘) < 𝑥)
106		rabn0 4394	. . . . . . . . . . . . . . . . . . 19 ⊢ ({𝑛 ∈ ℤ ∣ (𝑛 / 𝑘) < 𝑥} ≠ ∅ ↔ ∃𝑛 ∈ ℤ (𝑛 / 𝑘) < 𝑥)
107	105, 106	sylibr 234	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑥 ∈ ℝ ∧ 𝑘 ∈ ℕ) → {𝑛 ∈ ℤ ∣ (𝑛 / 𝑘) < 𝑥} ≠ ∅)
108	1	neeq1i 3002	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑇 ≠ ∅ ↔ {𝑛 ∈ ℤ ∣ (𝑛 / 𝑘) < 𝑥} ≠ ∅)
109	107, 108	sylibr 234	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑥 ∈ ℝ ∧ 𝑘 ∈ ℕ) → 𝑇 ≠ ∅)
110	1	reqabi 3456	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑛 ∈ 𝑇 ↔ (𝑛 ∈ ℤ ∧ (𝑛 / 𝑘) < 𝑥))
111	47	ad2antlr 727	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (((𝑥 ∈ ℝ ∧ 𝑘 ∈ ℕ) ∧ 𝑛 ∈ ℤ) → 𝑘 ∈ ℝ)
112	111, 100, 92	syl2anc 584	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((𝑥 ∈ ℝ ∧ 𝑘 ∈ ℕ) ∧ 𝑛 ∈ ℤ) → (𝑘 · 𝑥) ∈ ℝ)
113		ltle 11346	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝑛 ∈ ℝ ∧ (𝑘 · 𝑥) ∈ ℝ) → (𝑛 < (𝑘 · 𝑥) → 𝑛 ≤ (𝑘 · 𝑥)))
114	99, 112, 113	syl2anc 584	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝑥 ∈ ℝ ∧ 𝑘 ∈ ℕ) ∧ 𝑛 ∈ ℤ) → (𝑛 < (𝑘 · 𝑥) → 𝑛 ≤ (𝑘 · 𝑥)))
115	103, 114	sylbid 240	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝑥 ∈ ℝ ∧ 𝑘 ∈ ℕ) ∧ 𝑛 ∈ ℤ) → ((𝑛 / 𝑘) < 𝑥 → 𝑛 ≤ (𝑘 · 𝑥)))
116	115	impr 454	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝑥 ∈ ℝ ∧ 𝑘 ∈ ℕ) ∧ (𝑛 ∈ ℤ ∧ (𝑛 / 𝑘) < 𝑥)) → 𝑛 ≤ (𝑘 · 𝑥))
117	110, 116	sylan2b 594	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝑥 ∈ ℝ ∧ 𝑘 ∈ ℕ) ∧ 𝑛 ∈ 𝑇) → 𝑛 ≤ (𝑘 · 𝑥))
118	117	ralrimiva 3143	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑥 ∈ ℝ ∧ 𝑘 ∈ ℕ) → ∀𝑛 ∈ 𝑇 𝑛 ≤ (𝑘 · 𝑥))
119		breq2 5151	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑦 = (𝑘 · 𝑥) → (𝑛 ≤ 𝑦 ↔ 𝑛 ≤ (𝑘 · 𝑥)))
120	119	ralbidv 3175	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑦 = (𝑘 · 𝑥) → (∀𝑛 ∈ 𝑇 𝑛 ≤ 𝑦 ↔ ∀𝑛 ∈ 𝑇 𝑛 ≤ (𝑘 · 𝑥)))
121	120	rspcev 3621	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑘 · 𝑥) ∈ ℝ ∧ ∀𝑛 ∈ 𝑇 𝑛 ≤ (𝑘 · 𝑥)) → ∃𝑦 ∈ ℝ ∀𝑛 ∈ 𝑇 𝑛 ≤ 𝑦)
122	94, 118, 121	syl2anc 584	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑥 ∈ ℝ ∧ 𝑘 ∈ ℕ) → ∃𝑦 ∈ ℝ ∀𝑛 ∈ 𝑇 𝑛 ≤ 𝑦)
123	91, 109, 122	3jca 1127	. . . . . . . . . . . . . . . 16 ⊢ ((𝑥 ∈ ℝ ∧ 𝑘 ∈ ℕ) → (𝑇 ⊆ ℝ ∧ 𝑇 ≠ ∅ ∧ ∃𝑦 ∈ ℝ ∀𝑛 ∈ 𝑇 𝑛 ≤ 𝑦))
124		suprub 12226	. . . . . . . . . . . . . . . 16 ⊢ (((𝑇 ⊆ ℝ ∧ 𝑇 ≠ ∅ ∧ ∃𝑦 ∈ ℝ ∀𝑛 ∈ 𝑇 𝑛 ≤ 𝑦) ∧ (sup(𝑇, ℝ, < ) + 1) ∈ 𝑇) → (sup(𝑇, ℝ, < ) + 1) ≤ sup(𝑇, ℝ, < ))
125	123, 124	sylan 580	. . . . . . . . . . . . . . 15 ⊢ (((𝑥 ∈ ℝ ∧ 𝑘 ∈ ℕ) ∧ (sup(𝑇, ℝ, < ) + 1) ∈ 𝑇) → (sup(𝑇, ℝ, < ) + 1) ≤ sup(𝑇, ℝ, < ))
126	86, 125	syldan 591	. . . . . . . . . . . . . 14 ⊢ (((𝑥 ∈ ℝ ∧ 𝑘 ∈ ℕ) ∧ ((sup(𝑇, ℝ, < ) + 1) / 𝑘) < 𝑥) → (sup(𝑇, ℝ, < ) + 1) ≤ sup(𝑇, ℝ, < ))
127	126	ex 412	. . . . . . . . . . . . 13 ⊢ ((𝑥 ∈ ℝ ∧ 𝑘 ∈ ℕ) → (((sup(𝑇, ℝ, < ) + 1) / 𝑘) < 𝑥 → (sup(𝑇, ℝ, < ) + 1) ≤ sup(𝑇, ℝ, < )))
128	42	zcnd 12720	. . . . . . . . . . . . . . . 16 ⊢ ((𝑥 ∈ ℝ ∧ 𝑘 ∈ ℕ) → sup(𝑇, ℝ, < ) ∈ ℂ)
129		1cnd 11253	. . . . . . . . . . . . . . . 16 ⊢ ((𝑥 ∈ ℝ ∧ 𝑘 ∈ ℕ) → 1 ∈ ℂ)
130		nncn 12271	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑘 ∈ ℕ → 𝑘 ∈ ℂ)
131		nnne0 12297	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑘 ∈ ℕ → 𝑘 ≠ 0)
132	130, 131	jca 511	. . . . . . . . . . . . . . . . 17 ⊢ (𝑘 ∈ ℕ → (𝑘 ∈ ℂ ∧ 𝑘 ≠ 0))
133	132	adantl 481	. . . . . . . . . . . . . . . 16 ⊢ ((𝑥 ∈ ℝ ∧ 𝑘 ∈ ℕ) → (𝑘 ∈ ℂ ∧ 𝑘 ≠ 0))
134		divdir 11944	. . . . . . . . . . . . . . . 16 ⊢ ((sup(𝑇, ℝ, < ) ∈ ℂ ∧ 1 ∈ ℂ ∧ (𝑘 ∈ ℂ ∧ 𝑘 ≠ 0)) → ((sup(𝑇, ℝ, < ) + 1) / 𝑘) = ((sup(𝑇, ℝ, < ) / 𝑘) + (1 / 𝑘)))
135	128, 129, 133, 134	syl3anc 1370	. . . . . . . . . . . . . . 15 ⊢ ((𝑥 ∈ ℝ ∧ 𝑘 ∈ ℕ) → ((sup(𝑇, ℝ, < ) + 1) / 𝑘) = ((sup(𝑇, ℝ, < ) / 𝑘) + (1 / 𝑘)))
136	3	mptex 7242	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑘 ∈ ℕ ↦ (sup(𝑇, ℝ, < ) / 𝑘)) ∈ V
137	2	fvmpt2 7026	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑥 ∈ ℝ ∧ (𝑘 ∈ ℕ ↦ (sup(𝑇, ℝ, < ) / 𝑘)) ∈ V) → (𝐹‘𝑥) = (𝑘 ∈ ℕ ↦ (sup(𝑇, ℝ, < ) / 𝑘)))
138	136, 137	mpan2 691	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑥 ∈ ℝ → (𝐹‘𝑥) = (𝑘 ∈ ℕ ↦ (sup(𝑇, ℝ, < ) / 𝑘)))
139	138	fveq1d 6908	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 ∈ ℝ → ((𝐹‘𝑥)‘𝑘) = ((𝑘 ∈ ℕ ↦ (sup(𝑇, ℝ, < ) / 𝑘))‘𝑘))
140		ovex 7463	. . . . . . . . . . . . . . . . . 18 ⊢ (sup(𝑇, ℝ, < ) / 𝑘) ∈ V
141		eqid 2734	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑘 ∈ ℕ ↦ (sup(𝑇, ℝ, < ) / 𝑘)) = (𝑘 ∈ ℕ ↦ (sup(𝑇, ℝ, < ) / 𝑘))
142	141	fvmpt2 7026	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑘 ∈ ℕ ∧ (sup(𝑇, ℝ, < ) / 𝑘) ∈ V) → ((𝑘 ∈ ℕ ↦ (sup(𝑇, ℝ, < ) / 𝑘))‘𝑘) = (sup(𝑇, ℝ, < ) / 𝑘))
143	140, 142	mpan2 691	. . . . . . . . . . . . . . . . 17 ⊢ (𝑘 ∈ ℕ → ((𝑘 ∈ ℕ ↦ (sup(𝑇, ℝ, < ) / 𝑘))‘𝑘) = (sup(𝑇, ℝ, < ) / 𝑘))
144	139, 143	sylan9eq 2794	. . . . . . . . . . . . . . . 16 ⊢ ((𝑥 ∈ ℝ ∧ 𝑘 ∈ ℕ) → ((𝐹‘𝑥)‘𝑘) = (sup(𝑇, ℝ, < ) / 𝑘))
145	144	oveq1d 7445	. . . . . . . . . . . . . . 15 ⊢ ((𝑥 ∈ ℝ ∧ 𝑘 ∈ ℕ) → (((𝐹‘𝑥)‘𝑘) + (1 / 𝑘)) = ((sup(𝑇, ℝ, < ) / 𝑘) + (1 / 𝑘)))
146	135, 145	eqtr4d 2777	. . . . . . . . . . . . . 14 ⊢ ((𝑥 ∈ ℝ ∧ 𝑘 ∈ ℕ) → ((sup(𝑇, ℝ, < ) + 1) / 𝑘) = (((𝐹‘𝑥)‘𝑘) + (1 / 𝑘)))
147	146	breq1d 5157	. . . . . . . . . . . . 13 ⊢ ((𝑥 ∈ ℝ ∧ 𝑘 ∈ ℕ) → (((sup(𝑇, ℝ, < ) + 1) / 𝑘) < 𝑥 ↔ (((𝐹‘𝑥)‘𝑘) + (1 / 𝑘)) < 𝑥))
148	81	zred 12719	. . . . . . . . . . . . . 14 ⊢ ((𝑥 ∈ ℝ ∧ 𝑘 ∈ ℕ) → (sup(𝑇, ℝ, < ) + 1) ∈ ℝ)
149	148, 43	lenltd 11404	. . . . . . . . . . . . 13 ⊢ ((𝑥 ∈ ℝ ∧ 𝑘 ∈ ℕ) → ((sup(𝑇, ℝ, < ) + 1) ≤ sup(𝑇, ℝ, < ) ↔ ¬ sup(𝑇, ℝ, < ) < (sup(𝑇, ℝ, < ) + 1)))
150	127, 147, 149	3imtr3d 293	. . . . . . . . . . . 12 ⊢ ((𝑥 ∈ ℝ ∧ 𝑘 ∈ ℕ) → ((((𝐹‘𝑥)‘𝑘) + (1 / 𝑘)) < 𝑥 → ¬ sup(𝑇, ℝ, < ) < (sup(𝑇, ℝ, < ) + 1)))
151	150	adantlr 715	. . . . . . . . . . 11 ⊢ (((𝑥 ∈ ℝ ∧ sup(ran (𝐹‘𝑥), ℝ, < ) < 𝑥) ∧ 𝑘 ∈ ℕ) → ((((𝐹‘𝑥)‘𝑘) + (1 / 𝑘)) < 𝑥 → ¬ sup(𝑇, ℝ, < ) < (sup(𝑇, ℝ, < ) + 1)))
152	80, 151	syld 47	. . . . . . . . . 10 ⊢ (((𝑥 ∈ ℝ ∧ sup(ran (𝐹‘𝑥), ℝ, < ) < 𝑥) ∧ 𝑘 ∈ ℕ) → ((1 / (𝑥 − sup(ran (𝐹‘𝑥), ℝ, < ))) < 𝑘 → ¬ sup(𝑇, ℝ, < ) < (sup(𝑇, ℝ, < ) + 1)))
153	152	exp31 419	. . . . . . . . 9 ⊢ (𝑥 ∈ ℝ → (sup(ran (𝐹‘𝑥), ℝ, < ) < 𝑥 → (𝑘 ∈ ℕ → ((1 / (𝑥 − sup(ran (𝐹‘𝑥), ℝ, < ))) < 𝑘 → ¬ sup(𝑇, ℝ, < ) < (sup(𝑇, ℝ, < ) + 1)))))
154	153	com4l 92	. . . . . . . 8 ⊢ (sup(ran (𝐹‘𝑥), ℝ, < ) < 𝑥 → (𝑘 ∈ ℕ → ((1 / (𝑥 − sup(ran (𝐹‘𝑥), ℝ, < ))) < 𝑘 → (𝑥 ∈ ℝ → ¬ sup(𝑇, ℝ, < ) < (sup(𝑇, ℝ, < ) + 1)))))
155	154	com14 96	. . . . . . 7 ⊢ (𝑥 ∈ ℝ → (𝑘 ∈ ℕ → ((1 / (𝑥 − sup(ran (𝐹‘𝑥), ℝ, < ))) < 𝑘 → (sup(ran (𝐹‘𝑥), ℝ, < ) < 𝑥 → ¬ sup(𝑇, ℝ, < ) < (sup(𝑇, ℝ, < ) + 1)))))
156	155	3imp 1110	. . . . . 6 ⊢ ((𝑥 ∈ ℝ ∧ 𝑘 ∈ ℕ ∧ (1 / (𝑥 − sup(ran (𝐹‘𝑥), ℝ, < ))) < 𝑘) → (sup(ran (𝐹‘𝑥), ℝ, < ) < 𝑥 → ¬ sup(𝑇, ℝ, < ) < (sup(𝑇, ℝ, < ) + 1)))
157	45, 156	mt2d 136	. . . . 5 ⊢ ((𝑥 ∈ ℝ ∧ 𝑘 ∈ ℕ ∧ (1 / (𝑥 − sup(ran (𝐹‘𝑥), ℝ, < ))) < 𝑘) → ¬ sup(ran (𝐹‘𝑥), ℝ, < ) < 𝑥)
158	157	rexlimdv3a 3156	. . . 4 ⊢ (𝑥 ∈ ℝ → (∃𝑘 ∈ ℕ (1 / (𝑥 − sup(ran (𝐹‘𝑥), ℝ, < ))) < 𝑘 → ¬ sup(ran (𝐹‘𝑥), ℝ, < ) < 𝑥))
159	41, 158	syld 47	. . 3 ⊢ (𝑥 ∈ ℝ → (sup(ran (𝐹‘𝑥), ℝ, < ) < 𝑥 → ¬ sup(ran (𝐹‘𝑥), ℝ, < ) < 𝑥))
160	159	pm2.01d 190	. 2 ⊢ (𝑥 ∈ ℝ → ¬ sup(ran (𝐹‘𝑥), ℝ, < ) < 𝑥)
161		eqlelt 11345	. . 3 ⊢ ((sup(ran (𝐹‘𝑥), ℝ, < ) ∈ ℝ ∧ 𝑥 ∈ ℝ) → (sup(ran (𝐹‘𝑥), ℝ, < ) = 𝑥 ↔ (sup(ran (𝐹‘𝑥), ℝ, < ) ≤ 𝑥 ∧ ¬ sup(ran (𝐹‘𝑥), ℝ, < ) < 𝑥)))
162	30, 161	mpancom 688	. 2 ⊢ (𝑥 ∈ ℝ → (sup(ran (𝐹‘𝑥), ℝ, < ) = 𝑥 ↔ (sup(ran (𝐹‘𝑥), ℝ, < ) ≤ 𝑥 ∧ ¬ sup(ran (𝐹‘𝑥), ℝ, < ) < 𝑥)))
163	29, 160, 162	mpbir2and 713	1 ⊢ (𝑥 ∈ ℝ → sup(ran (𝐹‘𝑥), ℝ, < ) = 𝑥)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1086   = wceq 1536   ∈ wcel 2105   ≠ wne 2937  ∀wral 3058  ∃wrex 3067  {crab 3432  Vcvv 3477   ⊆ wss 3962  ∅c0 4338   class class class wbr 5147   ↦ cmpt 5230  dom cdm 5688  ran crn 5689   Fn wfn 6557  ⟶wf 6558  ‘cfv 6562  (class class class)co 7430   ↑_m cmap 8864  supcsup 9477  ℂcc 11150  ℝcr 11151  0cc0 11152  1c1 11153   + caddc 11155   · cmul 11157   < clt 11292   ≤ cle 11293   − cmin 11489   / cdiv 11917  ℕcn 12263  ℤcz 12610  ℚcq 12987

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-10 2138  ax-11 2154  ax-12 2174  ax-ext 2705  ax-rep 5284  ax-sep 5301  ax-nul 5311  ax-pow 5370  ax-pr 5437  ax-un 7753  ax-resscn 11209  ax-1cn 11210  ax-icn 11211  ax-addcl 11212  ax-addrcl 11213  ax-mulcl 11214  ax-mulrcl 11215  ax-mulcom 11216  ax-addass 11217  ax-mulass 11218  ax-distr 11219  ax-i2m1 11220  ax-1ne0 11221  ax-1rid 11222  ax-rnegex 11223  ax-rrecex 11224  ax-cnre 11225  ax-pre-lttri 11226  ax-pre-lttrn 11227  ax-pre-ltadd 11228  ax-pre-mulgt0 11229  ax-pre-sup 11230

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1539  df-fal 1549  df-ex 1776  df-nf 1780  df-sb 2062  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2726  df-clel 2813  df-nfc 2889  df-ne 2938  df-nel 3044  df-ral 3059  df-rex 3068  df-rmo 3377  df-reu 3378  df-rab 3433  df-v 3479  df-sbc 3791  df-csb 3908  df-dif 3965  df-un 3967  df-in 3969  df-ss 3979  df-pss 3982  df-nul 4339  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4912  df-iun 4997  df-br 5148  df-opab 5210  df-mpt 5231  df-tr 5265  df-id 5582  df-eprel 5588  df-po 5596  df-so 5597  df-fr 5640  df-we 5642  df-xp 5694  df-rel 5695  df-cnv 5696  df-co 5697  df-dm 5698  df-rn 5699  df-res 5700  df-ima 5701  df-pred 6322  df-ord 6388  df-on 6389  df-lim 6390  df-suc 6391  df-iota 6515  df-fun 6564  df-fn 6565  df-f 6566  df-f1 6567  df-fo 6568  df-f1o 6569  df-fv 6570  df-riota 7387  df-ov 7433  df-oprab 7434  df-mpo 7435  df-om 7887  df-1st 8012  df-2nd 8013  df-frecs 8304  df-wrecs 8335  df-recs 8409  df-rdg 8448  df-er 8743  df-map 8866  df-en 8984  df-dom 8985  df-sdom 8986  df-sup 9479  df-pnf 11294  df-mnf 11295  df-xr 11296  df-ltxr 11297  df-le 11298  df-sub 11491  df-neg 11492  df-div 11918  df-nn 12264  df-n0 12524  df-z 12611  df-q 12988

This theorem is referenced by:  rpnnen1lem6  13021